Hyperfiniteness and the Halmos-Rohlin theorem for nonsingular Abelian actions

Abstract:

Theorem 1. Let the countable abelian group $G$ act nonsingularly and aperiodically on Lebesgue space $(X,\mu )$. Then for each finite subset $A \subset G$ and $\varepsilon > 0\exists$ finite $B \subset G$ and $F \subset X$ with $\{ bF:b \in B\}$ disjoint and $\mu [({ \cap _{a \in A}}B - a)F] > 1 - \varepsilon$. Theorem 2. Every nonsingular action of a countable abelian group on a Lebesgue space is hyperfinite.